Depends entirely on the power output of the laser, the dispersal length and the size, weight and reflectivity of the sail. Since we are already talking about imaginary technology, any answer between "slower than current chemical rockets" and "some significant fraction of the speed of light" could be valid.
You explicitly asked for unknown materials. Because there is no known material because such a laser does not exist. We have solar sail prototypes that were using the sun's light, but even they are far off in terms of size/weight ratio and reflectivity.
No, I didn't ask for unknown materials; lasers can be built with known materials. Also solar sails aren't lightsails, and the topic here is lightsails.
